Electrons trapped in graphene magnetic quantum dots with mass term

Owing to the Klein tunneling phenomenon, the permanent confinement or localization of electrons within a graphene quantum dot is unattainable. Nonetheless, a constant magnetic field can transiently ensnare an electron within the quantum dot, giving rise to what are known as quasi-bound states characterized by finite lifetimes. To prolong the retention of electrons within the quantum dot, we introduce a mass term into the Hamiltonian, thereby inducing an energy gap. We resolve the Dirac equation to ascertain the eigenspinors, and by ensuring their continuity at the boundaries, we investigate the scattering behavior. Our findings indicate that the presence of an energy gap can extend the lifetimes of these quasi-bound states within the quantum dot. In particular, we demonstrate that even in the absence of a magnetic field, the scattering efficiency attains significant levels when the energy gap gets closed to the incident energy of an electron traversing the quantum dot. It is found that an augmentation in the electron density within the quantum dot results in an enhancement of the electron-trapping time.